How bridge beams bend |
A beam bends when it curves from an initially straight form.
If we now think of a bridge consisting almost entirely of beams, such as the Front Street Bridge in Canada
- shown in the picture to the right, then it will have just three chapters.
For any sizable bridge there has to be a network of beams working together - some along the length of the bridge and some across the width.
In essence beam bridges are really just very big manufactured complex planks. The main beams may span simply between two supports or continuously over several. The chapters for a simple footbridge, perhaps over a small stream, are relatively easy to see but they may not be totally distinct. 
For example the cross plank beams in the picture of the canal bridge (see right) are also the decking.
The handrails of a beam bridge are not normally part of the structure of the bridge but of course must have sufficient strength and robustness to resist anyone leaning on them.A major highway bridge is more complex but you can still spot the chapters. The best way is to stand underneath and look up at the underside of the structure. The picture of the Second Severn Bridge Crossing between England and Wales near Bristol (see left) 
shows the underside of the two massive concrete box beams. The bridge deck is usually a concrete slab or a steel reinforced plate. It may also act as the top flange of the main beam. On top of the bridge deck will be a wearing surface usually of asphalt on which the vehicles travel. The deck will support all the other normal street furniture such as drains, streetlights, crash and noise barriers and handrails. There are two basic forms of beam cross section shape - closed or open cross section. A closed cross section has a contained inner space like a rectangle or a triangle- a common example is a box beam or a tube. An open section has no contained inner space - a common example is an I beam. A very large beam is called a girder though the terms are often used interchangeably. What happens when a beam bends? Beams work mainly by transferring a flow of internal forces of bending moment and shear. You can get a feel for these forces when you bend over forwards or bend down. You sense your back being pulled in tension and your tummy being squashed in compression. Beams can also be axially compressed. The arch of a modern arch bridge as at the Salginatobel Bridge and the Clyde Arc Finnieston Bridge, is really a curved beam - so arranged that the bending forces are small. Beams have three degrees of freedom - movement in two directions and a rotation. Any restrained up-down movement creates shear, restrained longitudinal movement causes tension or compression and restrained rotation causes bending moment. To see bending with the naked eye we need something flexible like a strip of wood or plastic - a ruler will do the job perfectly.  Rest the ruler on two supports (two thick books will do nicely), one at each end. Now press down on the middle with a finger. The ruler will bend quite easily and you can see the curve of the bent shape as in the middle of the diagram on the right. The downward arrows are the forces of your finger pressing down and the upward arrows are the consequent reactions from the books resisting your finger. The harder you push the more the beam sags and the more the curvature of the bend. The bent ruler is an example of the simplest of all beam structures - engineers call it a simply supported beam for obvious reasons - it's just a beam sitting on uncomplicated supports If we insert an extra support at the centre of our simply supported beam then we will now have a beam with two spans and three supports. Engineers call that a continuous beam - a two span continuous beam. We can easily imagine creating 3 span, 4 span or even 5 span continuous beam by having the appropriate number of internal supports. We are making three very important assumptions. So what is going on inside the beam as it sags and hogs?  Let's make an imaginary cut through the centre of the simply supported beam. Indeed let's make two cuts to expose a small piece or element of the beam as at the top (a) of the left hand diagram. At each cut there are two internal forces required to maintain equilibrium. The first one is S, a vertical internal shear force needed to keep the vertical forces on any piece of the beam in balance. The second internal force is M and is a force called a bending moment that is needed to balance the tendency of any piece of the beam to turn or rotate. As the beam sags, each little element on the bottom of the beam is being pulled or stretched - in tension. At the same time each element at the top of the beam is being squashed or compressed. The stresses in the cross section change from tension at the bottom to compression at the top. These changes occur through the depth of the cross section - there is no variation across the width. As they change through the depth there is a point where the stress changes from tension to compression. At that point the stress is zero. We call it the 'neutral axis' of the cross section. Stresses are vectors (i.e.they have size and direction) and hence can be represented by arrows of a chosen length. In Part (b) of the diagram the compression stresses at each level through the top of the depth of the beam are depicted by an arrow pointing into the beam face at each level. Similarly the tensile stresses in the lower half of the beam are arrows pointing away. I have omitted the vertical shear forces to simplify the diagram. The internal forces change along the length of the beam because the degree of bending - the curvature - changes along the length of the beam. The total force due to the stresses above the neutral axis must balance those below. This must be so to maintain horizontal equilibrium. However because one is tension and the other is compression they create a turning effect. In Part (c) of the diagram I have replaced the distributed stresses above the neutral axis with a single arrow. This arrow represents the sum of all of the distributed stresses as a total force in compression in the top half of the cross section. Likewise the lower arrow represents the total force in tension in the bottom half. The turning effect is now clear as the top force pushes into the cross section and the bottom force pulls it out. With these internal forces in place every little piece of material in the beam would rotate - if it were free to do so - but it can't because the rest of the beam restrains it. Each element rotates but not as much as it would if it were entirely free. The internal force that creates this restraint is called the bending moment. Conventionally the two forces resisting the turning are shown using the curved arrows M in Part (a) and (d). To present the complete picture I have also reintroduced the shear force in (d). There are two types of bending moment and shear force. This is what we have been discussing so far. The values depend on the shape of the beam and the material from which it is made. The resisting moment and shear force must be bigger than the applied moment and shear force if the beam is to be safe. You will find more detail in the book. |
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